1. Area and Co-Area Formula 1.1. Hausdorff Measure. in This Section

1. Area and co-area formula

1.1. Hausdorff measure. In this section we will recall the definition of the Hausdorff measure and we will state some of its basic properties. A more detailed discussion is postponed to Section ??.

s/2 s Let ωs = π /Γ(1 + 2 ), s ≥ 0. If s = n is a positive integer, then ωn is volume of the 1 unit ball in Rn. Let X be a metric space. For ε > 0 and E ⊂ X we deﬁne ∞ ωs X Hs(E) = inf (diam A )s ε 2s i i=1 where the inﬁmum is taken over all possible coverings ∞ [ E ⊂ Ai with diam Ai ≤ ε. i=1 s Since the function ε 7→ Hε(E) is nonincreasing, the limit

s s H (E) = lim Hε(E) ε→0 exists. Hs is called the Hausdorﬀ measure. It is easy to see that if s = 0, H0 is the counting measure.

s P∞ s The Hausdorff content H∞(E) is defined as the infimum of i=1 ri over all coverings ∞ [ E ⊂ B(xi, ri) i=1 s of E by balls of radii ri. It is an easy exercise to show that H (E) = 0 if and only if s H∞(E) = 0. Often it is easier to use the Hausdorff content to show that the Hausdorff measure of a set is zero, because one does not have to worry about the diameters of the sets in the covering. The Hausdorff content is an outer measure, but very few sets are s actually measurable, and it is not countably additive on Borel sets. This is why H∞ is called content, but not measure.

Theorem 1.1. Hs is a metric outer measure i.e. Hs(E ∪ F ) = Hs(E) + Hs(F ) whenever E and F are arbitrary sets with dist (E,F ) > 0. Hence all Borel sets are Hs measurable.

It is an easy exercise to prove that Hs is an outer measure. The fact that it is a metric outer measure follows from the observation that if ε < dist (E,F )/2, we can assume that

1 n ωn n s If B ⊂ R is a ball, then 2n (diam B) = |B|. This explains the choice of the coefficient ωs/2 in the definition of the Hausdorff measure. 1 2 sets of diameter less than ε that cover E are disjoint from the sets of diameter less than ε that cover F . We leave details as an exercise. Finally measurability of Borel sets is a general property of metric outer measures.

The next result is very important and diﬃcult. We will prove it in Section ??

Theorem 1.2. Hn on Rn coincides with the outer Lebesgue measure Ln. Hence a set is Hn measurable if and only if it is Lebesgue measurable and both measures are equal on the class of measurable sets.

This result generalizes to the case of the Lebesgue measure on submanifolds of Rn. We will discuss it in the Subsection 1.3.

In what follows we will often use the Hausdorﬀ measure notation to denote the Lebesgue measure.

Proposition 1.3. If f : X ⊃ E → Y is a Lipschitz mapping between metric spaces, then Hs(f(E)) ≤ LsHs(E). In particular if Hs(E) = 0, then Hs(f(E)) = 0.

This is very easy. Indeed if A ⊂ E, then f(A) has diameter less than or equal to Ldiam A, where L is the Lipschitz constant of f. This observation and the deﬁnition of the Hausdorﬀ measure easily yields the result.

In particular, if f : Rn ⊃ E → Rm is a Lipschitz mapping and |E| = 0, then Hn(f(E)) = 0. We will prove a stronger result which is known as the Sard theorem. A more general version of the Sard theorem will be discussed in Section ??.

Theorem 1.4 (Sard). Let f : Rn ⊃ E → Rm be Lipschitz continuous and let

Crit (f) = {x ∈ E : rank apDf(x) < n}, then Hn(f(Crit (f))) = 0.

In the proof we will need the so called 5r-covering lemma. It is also called a Vitali type covering lemma. Here and in what follows by σB we denote a ball concentric with the ball B and σ times the radius.

Theorem 1.5 (5r-covering lemma). Let B be a family of balls in a metric space such that sup{diam B : B ∈ B} < ∞. Then there is a subfamily of pairwise disjoint balls B0 ⊂ B 3 such that [ [ B ⊂ 5B. B∈B B∈B0 If the metric space is separable, then the family B0 is countable and we can arrange it as a 0 ∞ sequence B = {Bi}i=1, so ∞ [ [ B ⊂ 5Bi . B∈B i=1 Remark 1.6. Here B can be either a family of open balls or closed balls. In both cases the proof is the same.

Proof. Let sup{diam B : B ∈ B} = R < ∞. Divide the family B according to the diameter of the balls R R F = {B ∈ B : < diam B ≤ } . j 2j 2j−1 S∞ Clearly B = j=1 Fj. Deﬁne B1 ⊂ F1 to be the maximal family of pairwise disjoint balls.

Suppose the families B1,..., Bj−1 are already deﬁned. Then we deﬁne Bj to be the maximal family of pairwise disjoint balls in

j−1 0 0 [ Fj ∩ {B : B ∩ B = ∅ for all B ∈ Bi} . i=1 0 S∞ Next we deﬁne B = j=1 Bj. Observe that every ball B ∈ Fj intersects with a ball in Sj Sj i=1 Bj. Suppose that B ∩ B1 6= ∅, B1 ∈ i=1 Bi. Then R R diam B ≤ = 2 · ≤ 2 diam B 2j−1 2j 1 and hence B ⊂ 5B1. The proof is complete.

Proof of the Sard theorem. Using the McShane extension (Theorem ??) we can assume that f is deﬁned on all of Rn and replace the approximate derivative by the classical one. Indeed, the set of points in E where the approximate derivative exists, but the extension to Rn is not diﬀerentiable at these points has measure zero and this set is mapped onto a set of Hn measure zero.

Let Z be the set of points in Rn such that Df(x) exists and rank Df(x) < n. We need to show that Hn(f(Z)) = 0. By splitting Z into bounded pieces we may assume that Z is 2 contained in the interior of the unit cube Q. For L > ε > 0 and x ∈ Z there is rx > 0

2Indeed, if each bounded piece of Z is mapped into a set of Hn measure zero, then Z is mapped into a set of measure zero. 4 such that B(x, rx) ⊂ Q and

|f(y) − f(x) − Df(x)(y − x)| < εrx if y ∈ B(x, 5rx).

Hence

dist (f(y),Wx) ≤ εrx for y ∈ B(x, 5rx),

n where Wx = f(x) + Df(x)(R ) is an aﬃne space through f(x). Clearly dim Wx ≤ n − 1. Thus

(1.1) f(B(x, 5rx)) ⊂ B(f(x), 5Lrx) ∩ {z : dist (z, Wx) ≤ εrx}.

Since dim Wx = k ≤ n − 1 we have that

n n−1 n H∞(f(B(x, 5rx)) ≤ CεL rx , where the constant C depends on n only. Indeed, the k dimensional ball B(f(x), 5Lrx)∩Wx can be covered by Lr k Ln−1 C x ≤ C εrx ε 3 balls of radius εrx. Then balls with radii 2εrx and the same centers cover the right hand side of (1.1). Thus

Ln−1 Hn (f(B(x, 5r )) ≤ C (4εr )n = C0εrnLn−1. ∞ x ε x x S From the covering Z ⊂ x∈Z B(x, rx) we can select a family of pairwise disjoint balls S B(xi, rxi ), i = 1, 2,... such that Z ⊂ i B(xi, 5rxi ). We have ∞ ∞ X X Hn (f(Z)) ≤ Hn (f(B(x , 5r )) ≤ CεLn−1 rn ≤ C0εLn−1, ∞ ∞ i xi xi i=1 i=1

because the balls B(xi, rxi ) are disjoint and contained in the unit cube; hence the sum of their volumes is less than one. Since ε can be arbitrarily small we conclude that n n H∞(f(Z)) = 0 and thus H (f(Z)) = 0.

Exercise 1.7. Show that if

•Hs(E) < ∞, then Ht(E) = 0 for all t > s ≥ 0; •Hs(E) > 0, then Ht(E) = ∞ for all 0 ≤ t < s.

3 and hence diameter 4εrx 5

Definition 1.8. The Hausdorff dimension is defined as follows. If Hs(E) > 0 for all s ≥ 0, then dimH (E) = ∞. Otherwise we define

s dimH (E) = inf{s ≥ 0 : H (E) = 0}.

It follows from the exercise that there is s ∈ [0, ∞] such that Ht(E) = 0 for t > s and Ht(E) = ∞ for 0 < t < s. Hausdorﬀ dimension of E equals s. It also easily follows from Proposition 1.3 that Lipschitz mappings do not increase the Hausdorﬀ dimension.

1.2. Countably rectiﬁable sets.

Definition 1.9. We say that a metric space X is countably n-rectifiable if there is a family n of Lipschitz mappings fi : R ⊃ Ei → X defined on measurable sets such that

∞ ! n [ H X \ f(Ei) = 0. i=1

In particular we can talk about sets X ⊂ Rm that are countably n-rectiﬁable.

Clearly any Borel subset of a countably n-rectiﬁable set is countably n-rectiﬁable.

In other words X is countably n-rectifiable if it can be covered by countably many Lipschitz images of subsets of Rn up to a set of Hn measure zero. Since Lipschitz mappings map sets of finite Hn measure onto sets of finite Hn measure, the Hn measure on X is

σ-ﬁnite and hence dimH X ≤ n. We do not require the mappings fi to be one-to-one and one can imagine that X can be very complicated. However as we will see, if X is a subset of Rm its structure is relatively simple.

Theorem 1.10. A Borel set E ⊂ Rm is countably n-rectiﬁable, m ≥ n, if and only if there 1 ∞ m is a sequence of n-dimensional C -submanifolds {Mi}i=1 of R such that

∞ ! n [ (1.2) H E \ Mi = 0. i=1

Proof. Clearly the condition (1.2) is suﬃcient for the countable n-rectiﬁability and we need m 1 to prove its necessity. Each mapping fi : Ei → R can be approximated by C -mappings 1 in the sense of Theorem ??(d). Using a sequence of such C maps we can approximate fi up to a set of measure zero. Since sets of measure zero are mapped by Lipschitz maps to 6

n m 1 sets of measure zero, we can simply assume that the mappings fi : R → R are C and ∞ ! n [ n H E \ fi(R ) = 0. i=1 n 1 A neighborhood of any point in R where rank Dfi = n is mapped to a C -submanifold of m n R and the remaining set of points where rank Dfi < n in mapped to a set of H measure zero by Theorem 1.4.

Deﬁnition 1.11. We say that a measurable mappings f : Rn ⊃ Ω → Rn, has the Lusin property N if for any measurable set A ⊂ Ω we have

|A| = 0 ⇒ |f(A)| = 0.

More generally we say that a measurable mapping f : Rn ⊃ A → X to a metric space has the Lusin property N if for any measurable set E ⊂ A we have

|A| = 0 ⇒ Hn(f(E)) = 0.

Exercise 1.12. Prove that a measurable mapping f : Rn ⊃ Ω → Rn maps Lebesgue measurable sets onto Lebesgue measurable sets if and only if it has the Lusin property N.

For example Lipschitz mappings have the Lusin property, Proposition 1.3.

Theorem 1.13. Let f : Rn ⊃ E → Rm be an a.e. approximately diﬀerentiable mapping with the Lusin property N, and let

Crit (f) = {x ∈ E : rank apDf(x) < n}, then Hn(f(Crit (f))) = 0.

Indeed, this result is a direct consequence of Lemma ??, the Sard theorem, and the Lusin property of f. A similar argument yields

Proposition 1.14. X ⊂ Rm, m ≥ n is countably n-rectiﬁable if and only if there are a.e. n m approximately diﬀerentiable mappings fi : R ⊃ Ei → R with the Lusin property N such that ∞ ! n [ H X \ f(Ei) = 0. i=1

Proposition 1.15. E ⊂ Rm, m ≥ n is countably n-rectiﬁable if and only if there is a locally Lipschitz map f : Rn → Rm such that Hn(E \ f(Rn)) = 0. 7

Indeed, we can assume in the definition of a countably n-rectifiable set that the sets Ei are contained in a unit cube. We can place such sets in disjoint unit cubes in Rn that are separated by a positive distance. On each cube we apply the McShane extension to the m mapping f : Ei → R . Then we glue the mappings to form a locally Lipschitz mapping n m 4 f : R → R by multiplying the extension of fi by a cut-off function that equals 1 on the unit cube that contains Ei. Note that the result is not true for countably rectifiable subsets of metric spaces, because in such a general setting the McShane theorem is not available.

1.3. The area formula. Recall the classical change of variable s formula.

Theorem 1.16. Let Φ:Ω → Rn be a C1 diﬀeomorphism between domains Ω ⊂ Rn and n 1 Φ(Ω) ⊂ R . If f :Ω → [0, ∞] is a nonnegative measurable function or if f|JΦ| ∈ L (Ω) is integrable, then Z Z −1 f(Φ (y)) dy = f(x)|JΦ(x)| dx, Φ(Ω) Ω where JΦ(x) = det DΦ(x) is the Jacobian of the diﬀeomorphism Φ.

In the case in which the function f is deﬁned on Φ(Ω) we have Z Z f(y) dy = (f ◦ Φ)(x)|JΦ(x)| dx, Φ(Ω) Ω where we assume that f ≥ 0 or that f ∈ L1(Ω). Theorem 1.16 generalizes to the case of integration over an n-dimensional submanifold M of Rm, m ≥ n. A neighborhood of any point in M can be represented as the image of a parametrization. Recall that a parametrization of M is a one-to-one mapping

n m Φ: R ⊃ Ω → R , Φ(Ω) ⊂ M of class C1 such that rank Dg(x) = n for all x ∈ Ω.

T Observe that det(DΦ) DΦ is the Gramm determinant of vectors ∂Φ/∂xi and hence pdet(DΦ)T DΦ(x) is the n-dimensional volume of the parallelepiped with edges

∂Φ(x)/∂xi. Thus it is natural to deﬁne

p T |JΦ(x)| = det(DΦ) DΦ(x), even if m > n. Note that this deﬁnition is consistent with the standard deﬁnition of the absolute value of the Jacobian when m = n.

4 We multiply each component of the function fi by a cut-oﬀ function. 8

In the case when m = n and Φ is a diffeomerphism, the change of variables formula implies that the Lebesgue measure |E| of a set E ⊂ Φ(Ω) equals Z |E| = |JΦ(x)| dx. Φ−1(E) If Φ : Rn ⊃ Ω → M ⊂ Rm is a parametrization, we define the Lebesgue measure (surface measure) σ(E) of a set E ⊂ Φ(Ω) by the formula (1.3), i.e. Z (1.3) σ(E) = |JΦ(x)| dx. Φ−1(E) If E ⊂ M is not necessarily contained in the image of a single parametrization, we divide it into small pieces that are contained in the images of parametrizations and we add the measures. One only needs to observe that the measure of a set does not depend on the choice of a parametrization. Indeed, suppose that E ⊂ Φ1(Ω1) ∩ Φ2(Ω2). By taking −1 smaller domains we can assume that Φ1(Ω1) = Φ2(Ω2). Then Φ1 ◦ Φ2 :Ω2 → Ω1 is a diffeomorphism and the change of variables formula easily implies that Z Z

|JΦ1 (x)| dx = |JΦ2 (x)| dx. −1 −1 Φ1 (E) Φ2 (E) Note that the formula (1.3) can be written as Z Z f(y) dσ(y) = (f ◦ Φ)(x)|JΦ(x)| dx, Φ(Ω) Ω where f is the characteristic function of the set E. Since measurable functions can be approximated by simple functions which are linear combinations of characteristic functions, standard limiting procedure yields

Theorem 1.17. Let Φ: Rn ⊃ Ω → M ⊂ Rm, m ≥ n be a parametrization of an n dimensional submanifold M ⊂ Rm. If f : Φ(Ω) → [0, ∞] is a nonnegative measurable function or if f ∈ L1(Φ(Ω)) is integrable, then Z Z f(y) dσ(y) = (f ◦ Φ)(x)|JΦ(x)| dx. Φ(Ω) Ω

1 For f ≥ 0 on Ω and for f|JΦ| ∈ L (Ω) the change of variables formula takes the form Z Z −1 (1.4) (f ◦ Φ )(y) dσ(y) = f(x)|JΦ(x)| dx. Φ(Ω) Ω According to Theorem 1.2 the Lebesgue measure in Rn coincides with the Hausdorff measure Hn. One can prove that the surface measure on M also coincides with the Hausdorff measure Hn defined either with respect to the Euclidean metric of Rm restricted to M or with respect to the natural Riemannian metric on M. We will not prove this fact, but this 9 result should not be surprising; M is locally very well approximated by tangent spaces and this approximation allows one to deduce the result from Theorem 1.2. In particular in both theorems Theorem 1.16 and 1.17 we can replace dy and dσ(y) by dHn(y).

The purpose of this section is to prove a far reaching generalization the change of variables formula.

If Φ : Rn ⊃ E → Rm, m ≥ n is approximately diﬀerentiable a.e., then we can formally deﬁne the Jacobian of Φ at almost every point of E by

p T |JΦ(x)| = det(DΦ) DΦ(x)

Theorem 1.18 (Area formula). Let Φ: Rn ⊃ E → Rm, m ≥ n be approximately differentiable a.e. Then we can redefine it on a set of measure zero in such a way that the new mapping satisfies the Lusin property N. If Φ is approximately differentiable a.e., satisfies 1 the Lusin property N and f : E → [0, ∞] is measurable or f|JΦ| ∈ L (E), then   Z Z X m (1.5) f(x)|JΦ(x)| dx =  f(x) dH (y). E Φ(E) x∈Φ−1(y)

Here we do not assume that the mapping Φ is one-to-one and this is why we have the sum on the right hand side, just to compensate the fact that the point y is the image of every point x in the set Φ−1(y). Note that since H0 is the counting measure formula (1.5) can be rewritten as Z Z Z n 0 m f(x)|JΦ(x)| dH (x) = f(x) dH (x) dH (y) . E Φ(E) Φ−1(y) The reason why we want to write it this way will be apparent when we will discuss the co-area formula.

Proof. Lemma ?? shows that away from a set Z of measure zero Φ has the Lusin property since it consists of Lipschitz pieces. Now if we modify Φ on the set Z and send the set to a single point, a new mapping Φ˜ will have the Lusin property and it will be equal to Φ almost everywhere. This proves the ﬁrst part of the theorem. Assume now that Φ has the Lusin property. Note that if we remove from E a subset of measure zero both sides of (1.5) will not change its value. It is obvious for the left hand side, but regarding the right hand side it follows from the Lusin property of Φ. We can also remove the subset of

E where JΦ = 0. According to Theorem 1.13, Φ maps this set onto a set of measure zero and hence both sides of (1.5) will not change its value after such a removal. This combined 10 with Theorem ?? allows us to assume that there are disjoint subsets Ei ⊂ E such that S 1 n m i Ei = E and C mappings Φi : R → R such that Φi = Φ on Ei, DΦi = apDΦ on Ei, rank DΦi = n on Ei. Dividing the sets into small pieces, if necessary, we can also assume that Φi is one-to-one in an open set containing Ei, i.e. it is a parametrization of an n-dimensional submanifold of Rm on that open set. According to the classical change of variables formula (1.4) we have5 Z Z −1 n f(x)|JΦi (x)| dx = f(Φi (y)) dH (y) Ei Φi(Ei) which yields Z ∞ Z ∞ Z X X −1 n f(x)|JΦ(x)| dx = f(x)|JΦi (x)| dx = f(Φi (y)) dH (y) E i=1 Ei i=1 Φi(Ei)   Z X n =  f(x) dH (y). Φ(E) x∈Φ−1(y) Indeed, if f ≥ 0 we can change the order of integration ans summation by the monotone convergence theorem. In the case of f ∈ L1 we consider separately the positive and negative parts of f.

Remark 1.19. It is necessary to require that Φ has the Lusin property. Indeed, if Φ maps a set of measure zero onto a set of positive measure, and f = 1, then the left hand side of the formula in Theorem 1.18 equals zero, but the right hand side is positive.

If f is a measurable function on Rm, and Φ : E → Rm is approximately diﬀerentiable a.e. and has the Lusin property N, then Theorem 1.18 applies to the function f ◦ Φ which is deﬁned on E. Note that f ◦ Φ is constant on the set Φ−1(y) and hence the area formula takes the form Z Z n (f ◦ Φ)(x)|JΦ(x)| dx = f(y)NΦ(y, E) dH (y), E Φ(E) where −1 NΦ(y, E) = #(Φ (y) ∩ E)

−1 is the cardinality of the set Φ (y)∩E. The function NΦ(·,E) is called the Banach indicatrix of Φ. This formula is true under the assumption that f ≥ 0 or under the integrability assumption of (f ◦ Φ)|JΦ|. More precisely if (f ◦ Φ)(x)|JΦ(x)| is integrable on E or if

5 We replace f in (1.4) by fχEi . 11 f(y)NΦ(y, E) is integrable on Φ(E), then the other function is integrable too and the formula is true.

Remark 1.20. Suppose that Φ : Q → Q is a homeomorphism of the unit cube Q = [0, 1]n that is identity on the boundary of the cube. Assume also that Φ is approximately diﬀerentiable a.e. and has the Lusin property. In this case the change of variables formula R shows that Q |JΦ| = 1. Since Φ is an orientation preserving homeomorphism, is it true that JΦ ≥ 0 a.e.? Surprisingly, one can ﬁnd such a homeomorphism with the property that

JΦ = −1 a.e. or that it is positive on a subset of a cube and negative on another subset and it is reasonable to conjecture that the only real constrain for a construction of a mapping

Φ with prescribed Jacobian is the condition that integral of |JΦ| over the cube equals one.

The area formula generalizes to the case of mappings between Riemannian manifolds. Submanifolds of Euclidean spaces are examples of Riemannian manifolds.

Theorem 1.21. The statement of Theorem 1.18 remains true if we replace Rn and Rm by n-dimensional and m-dimansional Riemannian manifolds respectively.

1.4. The co-area formula. The area formula is a generalization of the change of variable formula to the case of mappings from Rn to Rm, where m ≥ n. Surprisingly, it is also possible to generalize to change of variables formula to the case when m ≤ n; this is so called the co-area formula. First we need to generalize the Jacobian to the case of mappings Φ: Rn → Rm, m ≤ n. Suppose that Φ is diﬀerentiable at x ∈ Rn. If m < n, then (1.6) pdet(DΦ)T (DΦ)(x) = 0 because this is a formula for the n-dimensional volume of a parallelepiped which in our situation has the dimension ≤ m < n. That means (1.6) is not a good notion of the Jacobian whenm < n. Assume that rank of DΦ(x) is maximal, i.e. rank DΦ(x) = m ≤ n. If B is a n ball in the tangent space TxR centered at the origin, then DΦ(x)(B) is a non-degenerate m m-dimensional ellipsoid in TΦ(x)R . The kernel ker DΦ(x) is an n − m dimensional linear n subspace of TxR and DΦ(x) is a composition of two mappings; ﬁrst we take the orthogonal n ⊥ projection of TxR onto the m-dimensional space (ker DΦ(x)) and then we compose it with the linear isomorphism of m-dimensional spaces

⊥ m (1.7) DΦ(x)|(ker DΦ(x))⊥ : (ker DΦ(x)) → TΦ(x)R .

Now we deﬁne |JΦ(x)| as the absolute value of the Jacobain of the mapping (1.7), i.e. |JΦ(x)| is factor by which the linear mapping (1.7) changes volume. Geometrically speaking the 12 ellipsoid DΦ(x)(B) is the image of the m-dimensional ball B ∩ (ker DΦ(x))⊥ and hence Hm(DΦ(x)(B)) |J (x)| = . Φ Hm(B ∩ (ker DΦ(x))⊥)

If rank DΦ(x) < n we set |JΦ(x)| = 0. Although we deﬁned the Jacobian in geometric terms, there is a simple algebraic formula for |JΦ(x)| which follows from the polar decomposition of the linear mapping DΦ(x).

p T |JΦ(x)| = det(DΦ)(DΦ) (x).

Note that this is not the same formula as (1.6). However, the two formulas give the same value when m = n.

Exercise 1.22. Prove this formula using the polar decomposition of DΦ(x).

There is one more geometric interpretation of |JΦ(x)| when m ≤ n which easily follows from our geometric deﬁnition. Namely |JΦ(x)| equals the supremum of m-dimensional measures of all ellipsoids DΦ(x)(B), where the supremum is over all m-dimensional balls n B in TxR of volume 1. This reminds us of the the geometric interpretation of the length of the gradient of a real valued function as the maximal rate of of change of a function. The function has maximal growth in the direction the gradient which is orthogonal to det DΦ. In our case the maximal growth of the m-dimensional measure of m-dimensional balls in n TxR is also in the direction orthogonal to ker DΦ(x), see (1.7). This is the right intuition. If Φ : Rn → R, i.e. m = 1, one can easily see that

p T |JΦ(x)| = det(DΦ)(DΦ) (x) = |∇Φ(x)|.

Now we can state the co-area formula. We will actually state both area and co-area formula in one theorem, because it will help to see similarities and diﬀerences between the two formulas.

Theorem 1.23 (The area and the co-area formulas). Let Φ: Rn ⊃ E → Rm be a Lipschitz mapping deﬁned on a measurable set E ⊂ Rn. Let f ≥ 0 be a measurable function on E or 1 let f|JΦ| ∈ L (E). Then

• (Area formula) If n ≤ m, then Z Z Z n 0 m f(x)|JΦ(x)| dH (x) = f(x) dH (x) dH (y). E Rm Φ−1(y) 13

• (Co-area formula) If n ≥ m, then Z Z Z n n−m m f(x)|JΦ(x)| dH (x) = f(x) dH (x) dH (y). E Rm Φ−1(y)

We will not prove the co-area formula here, but we will show that it contains results like integration in the spherical coordinates and the Fubini theorem as special cases!

n Recall that if Φ : R → R, then |JΦ(x)| = |∇Φ(x)|. Taking Φ(x) = |x| we have |JΦ(x)| = 1 everywhere except at the origin. Since the image of Φ is [0, ∞), the co-area formula reads as Z Z ∞ Z f(x) dHn(x) = f(x) dHn−1(x) dr Rn 0 ∂B(0,r) which is the formula for the integration in the spherical coordinates.

Let now Φ : Rn → Rm, m < n be the projection on the ﬁrst m coordinates

Φ(x1, . . . , xn) = (x1, . . . , xm). Then |JΦ(x)| = 1 and we have Z Z Z n n−m m f(x) dH (x) = f(x1, . . . , xn) dH (xm+1, . . . , xn) dH (x1, . . . , xm) Rn Rm Rn−m which is the Fubini theorem.

n Let f : R → R be an arbitrary Lipschitz function. Taking Φ = f we have JΦ(x)| = |∇f(x)|; taking the function f in the co-area formula to be equal6 1 we have Z Z ∞ |∇f(x)| dx = Hn−1({f = t}) dx. Rn −∞ As an application of the co-area formula we will prove

Theorem 1.24. If Φ: Rn → Rm is Lipschitz continuous, then for a.e. y ∈ Rm, Φ−1(y) is countably (n − m)-rectiﬁable.

Proof. If m > n, then Hm(f(Rn)) = 0, so Φ−1(y) = ∅ for a.e. y ∈ Rn and the empty set is countably rectiﬁable. Thus we can assume that m ≤ n. Assume for a moment that Φ ∈ C1. Then according to the implicit function theorem

Φ−1(y) ∩ {rank DΦ = m} is a C1,(n − m)-dimensional submanifold of Rn and it follows from the co-area formula that n−m −1 H (Φ (y) ∩ {rank DΦ < m}) = 0 for Hm-a.e. y ∈ Rm.

6This might be slightly confusing since we have a double meaning of f. 14

Thus for almost all y,Φ−1 is a manifold plus a set of measure zero. Hence it is countably (n − m)-rectiﬁable. To prove the result in the case in which Φ is Lipschitz it suﬃces to use 1 Theorem ?? which reduces the problem to the C -case.

1.5. The Eilenberg inequality.

Deﬁnition 1.25. A metric space is said to be boundednly compact if bounded and closed sets are compact.

An important step in the proof of the co-area formula is the following

Theorem 1.26 (Eilenberg inequality). Let Φ: X → Y be a Lipschitz mapping between boundedly compact metric spaces. Let 0 ≤ m ≤ n be real numbers.7 Assume that E ⊂ X is Hn-measurable with Hn(E) < ∞. Then

(1) Φ−1(y) ∩ E is Hn−m-measurable for Hm-almost all y ∈ Y . (2) y 7→ Hn−m(Φ−1(y) ∩ E) is Hm-measurable.

Moreover Z ω ω Hn−m(Φ−1(y) ∩ E) dHm(y) ≤ (Lip (Φ))m m n−m Hn(E). Y ωn

Observe that the left hand side corresponds to the right hand side in the co-area formula m with f = 1. Observe also that |JΦ| can be estimated by Lip (Φ) , and then the integral of the Jacobian over E can be estimated from above by Lip (Φ)mHn(E). This shows a deep connection between the co-area formula and the Eilenberg inequality. Since we used the estimate from the above we only have an inequality and one cannot expect equality in the Eilenberg inequality. What is remarkable is that the Eilenberg inequality is true in a great generality of boundedly compact metric spaces where diﬀerentiable structure is not available. We will prove Theorem 1.26 under the additional assumption that X = Rn and Y = Rm.

The measurability of the function y 7→ Hn−m(Φ−1(y) ∩ E) is far from being obvious and we will want to integrate this function before proving its measurablility. To do this we will have to use the upper Lebesgue integral.

7Not necessarily integers. 15

Definition 1.27. For a nonnegative function f : X → [0, ∞] defined µ-a.e. on a measure space (X, µ) the upper Lebesgue integral is defined as Z ∗ Z f dµ = inf φ dµ : 0 ≤ f ≤ φ and φ is µ-measurable .

We do not assume measurability of f. Clearly if f is measurable the upper Lebesgue ineqgral equals the Lebesgue integral.

An important property of the upper integral is that if R ∗ f dµ = 0, then f = 0, µ-a.e. R and hence it is measurable. Indeed, there is a sequence 0 ≤ f ≤ φn such that φn dµ → 0. 1 That means φn → 0 in L (µ). Taking a subsequence we get φnk → 0, µ-a.e. which proves that f = 0, µ-a.e.

Proof of Theorem 1.26 when X = Rn and Y = Rm. For ever positive integer k > 0 there is a covering ∞ [ 1 E ⊂ A ,A is closed, diam A < ik ik ik k i=1 such that ∞ ωn X 1 (1.8) (diam A )n ≤ Hn(E) + . 2n ik k i=1 It follows directly from the deﬁnition of the Hausdorﬀ measure that ∞ n−m −1 ωn−m X −1 n−m (1.9) H (Φ (y) ∩ E) ≤ lim inf diam (Φ (y) ∩ Aik) . 2n−m k→∞ i=1 For any set A ⊂ X we have

−1 −1 n−m diam (Φ (y) ∩ A) = diam (Φ (y) ∩ A)χΦ(A)(y) ≤ (diam A) χΦ(A)(y). Hence (1.9) yields ∞ n−m −1 ωn−m X n−m H (Φ (y) ∩ E) ≤ lim inf (diam Aik) χ (y). 2n−m k→∞ Φ(Aik) i=1 The function on the right hand side is measurable. Hence Fatou’s lemma yields Z ∗ Z ∞ ωn−m X Hn−m(Φ−1(y) ∩ E) dHm(y) ≤ lim inf (diam A )n−mχ (y) dHm(y) n−m ik Φ(Aik) m 2 k→∞ m R R i=1 ∞ ωn−m X n−m m = lim inf (diam Aik) H (Φ(Aik)). 2n−m k→∞ i=1

If p ∈ Aik, then

Φ(Aik) ⊂ B(f(p), Lip (Φ) diam Aik) 16 and hence ω Hm(Φ(A )) ≤ m (Lip (Φ))m(diam A )m. ik 2m ik Thus Z ∗ n ∞ n−m −1 m ωn−m ωm 2 ωn X n H (Φ (y) ∩ E) dH (y) ≤ n−m m lim inf n (diam Ak) m 2 2 ω k→∞ 2 R n i=1 ω ω ≤ (Lip (Φ))m n−m n Hn(E), ωn by (1.8). It remains to prove the Hn−m-measurability of the sets Φ−1(y)∩E for Hm almost all y ∈ Rm and the Hm-measurability of the function ϕ(y) = Hn−m(Φ−1(y) ∩ E). Note that if Hn(E) = 0, then Hn−m(Φ−1(y) ∩ E) = 0 for Hm almost every y ∈ Rm by the upper integral estimate. This observations shows that we can ignore subsets of E of Hn measure zero. Thus we can assume that E is the union of an increasing sequence of compact sets S∞ −1 E k=1 Ek, Ek ⊂ Ek+1. Note that the sets Φ (y) ∩ Ek are compact for every y and hence Φ−1(y) ∩ E is Borel as the union of compact sets. Thus it suﬃces to prove that every n−m −1 function y 7→ H (Φ (y) ∩ Ek) is Borel measurable, because then the function

n−m −1 n−m −1 ϕ(y) = H (Φ (y) ∩ E) = lim H (Φ (y) ∩ Ek) k→∞ will also be Borel. Hence we can assume that E is compact. It remains to prove that for every t ∈ R the set

m n−m −1 (1.10) y ∈ R : H (Φ (y) ∩ E) ≤ t is Borel. If t < 0, then the set is empty, so we can assume that t ≥ 0. Since the set in (1.10) can be written as

m m n−m −1 (R \ Φ(E)) ∪ Φ(E) ∩ y ∈ R : H (Φ (y) ∩ E) ≤ t and Rm \ Φ(E) is open, it remains to prove that the set

m n−m −1 Φ(E) ∩ y ∈ R : H (Φ (y) ∩ E) ≤ t is Borel. In the definition of the Hausdorff measure we may restrict to coverings by open sets. However this family of sets is uncountable and we would like to have a countable family of sets from which we would choose coverings. Let F be the family of all open sets in Rn that are finite unions of balls with rational centers and radii. The family F is countable and we claim that it can be used as the family of sets from which we choose coverings provided we define the Hausdorff measure of a compact set K. Indeed, first we S cover the set K by open sets, K ⊂ i Ui Each open set is the union of a family of balls with rational centers and radii. These balls form a covering of K and hence we can select 17

SN 0 a ﬁnite subcovering K ⊂ j=1 Bj. Now we replace each set Ui by Ui which is the union S 0 of all the balls Bj that are contained in Ui. We obtain a new covering K ⊂ i Ui . Clearly 0 0 Ui ⊂ Ui and Ui ∈ F.

Let Fi be the collection of all ﬁnite families {Ui1,...,Uik} ⊂ F such that

k 1 ωn−m X 1 diam U < , j = 1, 2,..., and (diam U )n−m ≤ t + . ij i 2n−m ij i j=1

Note that each of the sets Uij is the union of a finite family of balls. The family Fi is also countable. Clearly we define this family to deal with the coverings of the set Φ−1(y) ∩ E that satisfies Hn−m(Φ−1(y) ∩ E) ≤ t.

If U ⊂ Rn is open, then

Φ(E) ∩ {y :Φ−1(y) ∩ E ⊂ U} = Φ(E) \ Φ(E \ U) is Borel, because both of the sets f(E) and f(E \ U) are compact. In particular the set

k !! [ [ Vi = Φ(E) \ Φ E \ Uij

{Ui1,...,Uik}∈Fi j=1 is Borel as a countable union over the entire family Fi. We will prove that

m n−m −1 \ (1.11) Φ(E) ∩ {y ∈ R : H (Φ (y) ∩ E) ≤ t} = Vi.

Clearly the set on the right hand side is Borel.

If y ∈ E and Hn−m(Φ−1(y) ∩ E) ≤ t, then for any i we can ﬁnd a covering

−1 Φ (y) ∩ E ⊂ Ui1 ∪ ... ∪ Uik, {Ui1,...,Uik} ∈ Fi.

Sk T∞ Thus y 6∈ Φ(E \ j=1 Uij) and hence y ∈ Vi. Since i can be chosen arbitrarily, y ∈ i=1 Vi. T∞ On the other hand if y ∈ i=1 Vi then y ∈ Φ(E) and for all i, y ∈ Vi, i.e. there is Sk −1 {Ui1,...,Uik} ∈ Fi such that y 6∈ Φ(E \ i=1 Uik), i.e. Φ (y) ∩ E ⊂ Ui1 ∪ ... ∪ Uik, so

1 Hn−m(Φ−1(y) ∩ E) ≤ t + . 1/i i

n−m −1 Taking the limit as i → ∞ we obtain H (Φ (y) ∩ E) ≤ t. The proof is complete. 18

1.6. Integral geometric measure. We say that a metric space is purely Hm-unrectifiable if for any Lipschitz mapping f : Rn ⊃ A → X we have Hm(f(A)) = 0. It easily follows from the definition that E ⊂ Rn is purely Hm-unrectifiable if and only if for any countably Hm rectifiable set F ⊂ Rn, Hm(E ∩ F ) = 0.

Theorem 1.28. If Hm(X) < ∞, then there is a Borel countably rectifiable set E ⊂ X such that X \ E is purely Hm-unrectifiable. Hence X has a decomposition into a rectifiable and a nonrectifiable parts X = E ∪ (X \ E). This decomposition is unique up to sets of Hm-measure zero.

Proof. Let M be the supremum of Hm(E) over all Hm-countably rectifiable Borel sets m m E ⊂ X. Hence there are Borel countably H -rectifiable sets Ei ⊂ X such that H (Ei) > S M − 1/i. It is easily to see that E = i Ei satisfies the claim of the theorem. Uniqueness is easy.

Definition 1.29. Let E ⊂ Rn be a Borel set and let 1 ≤ m ≤ n be integers. If m < n, the integral geometric measure Im of E is defined as Z Z m 1 m ∗ (1.12) I (E) = Np(y, E) dH (y) dϑn,m(p), β(n, m) p∈O∗(n,m) y∈Im p where O∗(n, m) is the space of orthogonal projections p from Rn onto m-dimensional n ∗ linear subspaces of R , Im p is the image of the projection and ϑn,m is the Haar measure on O∗(n, m) invariant under the action of O(n), normalized to have total mass 1. Moreover −1 Np(y, E) is the Banach indicatrix, i.e. Np(y, E) = #(p (y) ∩ E). The coefficient β(n, m) will be defined later. If m = n we simply define Im(E) = Hm(E).

Thus roughly speaking Im(E) is defined as follows. We fix an m-dimensional subspace of Rn and denote by p the orthogonal projection from Rn onto that subspace. Next we compute the measure of the projection of the set E onto that subspace taking into account the multiplicity function Np and then we average resulting measures over all possible ∗ ∗ projections p ∈ O (n, m). Note that since the measure ϑn,m is invariant under rotations m m n O(n), I (E1) = I (E2) if E1,E2 ⊂ R are isometric. We still need to define the coefficient β(n, m). Let [0, 1]m ⊂ Rm × Rn−m = Rn be the m-dimensional unit cube in Rn. We define Z Z m m ∗ β(n, m) = Np(y, [0, 1] ) dH (y) dϑn,m(p). p∈O∗(n,m) y∈Im p Clearly β(n, m) is a positive constant and with its definition Im([0, 1]m) = 1. Note that Im(Q) = Hm(Q) for any m-dimensional cube in Rn regardless how the cube is positioned 19

∗ in the space. This follows from the O(n) invariance of the measure ϑn,m and from the fact that both measures Im and Hm scale in the same way under homothetic transformations. Hence Im(E) = Hm(E) if E is an m-dimensional polyhedron in Rn. Indeed, up to a set of Hm-measure zero such a polyhedron is the union of countably many m-dimensional cubes and if Hm(A) = 0, then Im(A) = 0. This observation can be generalized to arbitrary countably Hm-dectiﬁable sets.

Theorem 1.30 (Federer). If E ⊂ Rn is countably Hm-rectiﬁable, m ≤ n, then Im(E) = Hm(E).

Proof. We can assume that m < n, because In = Hn by the definition. It suffices to assume that E is a subset of an m-dimensional C1-submanifold Mm ⊂ Rn. Indeed, the general case will follow from Theorem 1.10 and the fact that Hm(A) = 0 implies that Im(A) = 0. Let p0 be the restriction of p ∈ O∗(n, m) to Mm. the area formula yields Z Z m m (1.13) |Jp0 (x)| dH (x) = Np(y, E) dH (y). E Im p Let L be an m-dimensional affine subspace of Rn. Let p00 be the restriction of p to L 00 and let |Jp00 | be the Jacobian of the orthogonal projection p of L onto Im p. Clearly m n |Jp(x)| = |Jp00 |,where L = TxM is regarded as an affine subspace of R . Observe that Z ∗ |Jp00 | dϑn,m(p) = C(n, m) p∈O∗(n,m) ∗ is a constant that depends on n and m only. Indeed, the measure ϑn,m is invariant under rotations O(n) and hence we can rotate L without changing the value of the integral, so that L is parallel to Rm × {0} ⊂ Rn. Hence (1.13) yields Z Z m ∗ m Np(y, E) dH (y) dϑn,m(p) = C(n, m)H (E). p∈O∗(n,m) y∈Im p m Taking E = [0, 1] we see that C(n, m) = β(n, m).

The next result is a celebrated structure theorem of Besicovitch-Federer which we state without proof.

Theorem 1.31 (Structure theorem). If E ⊂ Rn, Hm(E) < ∞, m < n is purely Hm- unrectiﬁable, then Im(E) = 0.

Thus any set E ⊂ Rn with Hm(E) < ∞ can be decomposed into a rectiﬁable part on which Hm = Im and a non-rectiﬁable part on which Im = 0. This says a lot about 20 the structure of E, which explains the name of the theorem. This result also implies that Hm(E) ≥ Im(E) for any Borel set E ⊂ Rn.

One can construct Cantor type sets with Hm(E) > 0, but Im(E) = 0. Clearly E must be purely Hm-unrectiﬁable. However, the integral geometric measure can be used to detect the Hausdorﬀ dimension of a set:

m m Theorem 1.32 (Mattila). If dimH E > m, then I (E) = ∞. Hence I (E) < ∞ implies that dimH E ≤ m.

The proof requires quite a lot of harmonic analysis and potential theory and we will not present it here.

Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pitts- burgh, PA 15260, USA, [email protected]